### 5.3 Entropy of d-dimensional extreme black holes

The extremal black holes play a special role in gravitational theory. These black holes are characterized by vanishing Hawking temperature , which means that in the metric (165) the near-horizon expansion in the metric function starts with the quadratic term . Topologically, the true extremal geometry is different from the non-extremal one. Near the horizon the non-extremal static geometry looks like a product of a two-dimensional disk (in the plane ) and a -dimensional sphere. Then, the horizon is the center in the polar coordinate system on the disk. Contrary to this, an extremal geometry in the near-horizon limit is a product of a two-dimensional cylinder and a -dimensional sphere. Thus, the horizon in the extremal case is just another boundary rather than a regular inner point, as in the non-extremal geometry. However, one may consider a certain limiting procedure in which one approaches the extremal case staying all the time in the class of non-extremal geometries. This limiting procedure is what we shall call the “extremal limit”. A concrete procedure of this type was suggested by Zaslavsky [226]. One considers a sequence of non-extreme black holes in a cavity at and finds that there exists a set of data such that the limit is well defined. Even if one may have started with a rather general non-extremal metric, the limiting geometry is characterized by very few parameters. In this sense, one may talk about “universality” of the extremal limit. In fact, in the most interesting (and tractable) case the limiting geometry is the product of two-dimensional hyperbolic space with the -dimensional sphere. Since the limiting geometry belongs to the non-extreme class, its classical entropy is proportional to the horizon area in accord with the Bekenstein–Hawking formula. Then, the entanglement entropy of the limiting geometry is a one-loop quantum correction to the classical result. The universality we have just mentioned suggests that this correction possesses a universal behavior in the extreme limit and, since the limiting geometry is rather simple, the limiting entropy can be found explicitly. The latter was indeed shown by Mann and Solodukhin in [172].

#### 5.3.1 Universal extremal limit

Consider a static spherically-symmetric metric in the following form

where is the metric on the -dimensional unit sphere, describing a non-extreme hole with an outer horizon located at . However, the analysis can be made for a more general metric, in which , the limiting geometry is the simplest in the case we consider in Eq. (243). The function in Eq. (243) can be expanded as follows
It is convenient to consider the geodesic distance as a radial coordinate. Retaining the first two terms in Eq. (244), we find, for , that
In order to avoid the appearance of a conical singularity at , the Euclidean time in Eq. (243) must be compactified with period , which goes to infinity in the extreme limit . However, rescaling yields a new variable having period . Then, taking into account Eq. (245), one finds for the metric (243)
where we have introduced the variable . To obtain the extremal limit one just takes . The limiting geometry
is that of the direct product of a 2-dimensional space and a -sphere and is characterized by a pair of dimensional parameters and . The parameter sets the radius of the -dimensional sphere, while the parameter is the curvature radius for the 2-space. Clearly, this two-dimensional space is the negative constant curvature space . This is the universality we mentioned above: although the non-extreme geometry is in general described by an infinite number of parameters associated with the determining function , the geometry in the extreme limit depends only on two parameters and . Note that the coordinate is inadequate for describing the extremal limit (247) since the coordinate transformation (245) is singular when . The limiting metric (247) is characterized by a finite temperature, determined by the periodicity in angular coordinate .

The limiting geometry (247) is that of a direct product of 2d hyperbolic space with radius and a 2D sphere with radius . It is worth noting that the limiting geometry (247) precisely merges near the horizon with the geometry of the original metric (243) in the sense that all the curvature tensors for both metrics coincide. This is in contrast with, say, the situation in which the Rindler metric is considered to approximate the geometry of a non-extreme black hole: the curvatures of both spaces do not merge in general.

For a special type of extremal black holes , the limiting geometry is characterized by just one dimensionful parameter. This is the case for the Reissner–Nordström black hole in four dimensions. The limiting extreme geometry in this case is the well-known Bertotti–Robinson space characterized by just one parameter . This space has remarkable properties in the context of supergravity theory that are not the subject of the present review.

#### 5.3.2 Entanglement entropy in the extremal limit

Consider now a scalar field propagating on the background of the limiting geometry (247) and described by the operator

where is the Ricci scalar. For the metric (247) characterized by two dimensionful parameters and , one has that . For a -dimensional conformally-coupled scalar field we have . In this case

The calculation of the respective entanglement entropy goes along the same lines as before. First, one allows the coordinate , which plays the role of the Euclidean time, to have period . For the metric (247) then describes the space , where is the hyperbolic space coinciding with everywhere except the point , where it has a conical singularity with an angular deficit . The heat kernel of the Laplace operator on is given by the product

where and are the heat kernels, of the Laplace operator on and respectively. The effective action reads
where is a UV cut-off. On spaces with constant curvature the heat kernel function is known explicitly [34]. In particular, on a 2D space of negative constant curvature, the heat kernel has the following integral representation:
where . In Eq. (251) is the geodesic distance between the points on . Between two points and the geodesic distance is given by . The heat kernel on the conical hyperbolic space can be obtained from (251) by applying the Sommerfeld formula (22). Skipping the technical details, available in [172], let us just quote the result for the trace
where .

Let us denote the trace of the heat kernel of the Laplace operator on a -dimensional sphere of unit radius. The entanglement entropy in the extremal limit then takes the form [172, 206]

The function has the following small- expansion
The trace of the heat kernel on a sphere is known in some implicit form. However, for our purposes a representation in a form of an expansion is more useful,
where is the area of a unit radius sphere . The first few coefficients in this expansion can be calculated using the results collected in [219],

We shall consider some particular cases.

##### d=4.
The entanglement entropy in the extreme limit is

where is the UV finite part of the entropy. For minimal coupling this result was obtained in [172]. The first term in Eq. (257) is proportional to the horizon area , while the second term is a logarithmic correction to the area law. For conformal coupling , the logarithmic term is

##### d=5.
The entropy is

To simplify the expressions in higher dimensions we consider only the case of the conformal coupling .

The entropy takes the form:

##### d=8.

Two examples of the extreme geometry are of particular interest.

##### Entanglement entropy of the round sphere in Minkowski spacetime.
Consider a sphere of radius in flat Minkowski spacetime. One can choose a spherical coordinate system so that the surface is defined as and , and variables are the angular coordinates on . The -metric reads

where is a metric on sphere of unit radius. Metric (264) is conformal to the metric
which describes the product of two-dimensional hyperbolic space with coordinates and the sphere . Note that both spaces, and , have the same radius . Metric (265) describes the spacetime, which appears in the extremal limit of a -dimensional static black hole. In the hyperbolic space we can choose a polar coordinate system with its center at point ,
(for small one has that as in the polar system in flat spacetime) so that the metric takes the form
In this coordinate system the surface is defined by the condition . In the entanglement entropy of a conformally-coupled scalar field the logarithmic term is conformally invariant. Therefore, it is the same [206] for the entropy of a round sphere of radius in Minkowski spacetime and in the extreme limiting geometry (267). In various dimensions can be obtained from the results (258) – (262) by setting . One finds in , in and in . For the logarithmic term in the entropy of a round sphere has been calculated by Casini and Huerta [38] directly in Minkowski spacetime. They have obtained in all even dimensions up to . Subsequently, Dowker [72] has extended this result to and . In arbitrary dimension the logarithmic term can be expressed in terms of Bernulli numbers as is shown in [38] and [72]

##### Entanglement entropy of the extreme Reissner–Nordström black hole in d dimensions.
As was shown by Myers and Perry [180] a generalization of the Reissner–Nordström solution to higher dimension is given by Eq. (243) with

In the extreme limit . Expanding Eq. (268) near the horizon one finds, in this limit, that . Thus, this extreme geometry is characterized by values of radii and . In dimension we have as in the case considered above. In dimension the two radii are different, . For the conformal coupling the values of logarithmic term in various dimensions are presented in Table 1.

Table 1: Coefficients of the logarithmic term in the entanglement entropy of an extreme Reissner–Nordström black hole.
 4 6 8